## Math 284 Differential Equations (Spring 2021)

 Instructor Mark Pernarowski Textbook Differential Equations (9th ed.), Nagle, Saff, Snider Office Hours Schedule  (email to schedule an Online meeting) email pernarow @ montana.edu Lectures MWF 11:00-11:50am  (Face-Face Lewis 304) T        11:00-11:50am  (Online WeBeX) T        is just another lecture but online.

 Grading: The course % is determined by:   Midterm 1      M1           100    Midterm 2      M2           100   Final                F               100   Quizzes           Q             100  ________________________________                                           400          % = (M1+M2+F+HW)/4 The final is not comprehensive.Six quizzes each worth 20 pointswill be given. Your best 5 quizscores determine Q above. Exam and quiz scores will be posted in D2L. Exam and quiz dates are indicatedon the schedule below. Their contentwill be announced in class and posted below All exams/quizzes are closed book. No electronic devices/phones are permitted. Syllabus: Material covered in text is from:Chapter 1 Introductory DefinitionsChapter 2 First Order ODE MethodsChapter 3 First Order ModelsChapter 4 Second Order Linear ODE MethodsChapter 6 Higher Order Differential EquationsChapter 7 Laplace TransformsChapter 9 Linear Systems   Homework: Suggested homework is listed below.Although the homework is not gradedit is representative of the kinds ofquestions which will be on quizzesand exams.Additional review handouts and lecturenotes will be posted in D2L under "Contents" as the course develops. Covid Policy University Policy

### Schedule

Below is a calendar showing the schedule of quizzes (orange) and exams (red) and  holidays (green).
An approximate schedule of textbook section numbers is included and will be updated.

 Sunday Monday Tuesday Wed Thursday Friday Saturday 10 11       (1.1)Classes start 12        (1.2) 13       (1.2)/(2.1) 14 15      (2.2) 16 17 18 MLK Day 19        (2.2)/(2.3) 20        (2.3)/(2.4) 21 22       (2.4)Quiz 1 23 24 25       (2.6) 26        (2.6) 27       (2.6) 28 29        (3.2) 30 31 1        (3.2) 2          (3.4) 3         (3.5) 4 5.       (4.1)Quiz 2 6 7 8       (4.2) 9       (4.3) 10Review 11 12Midterm 1 13 14 15 Pres. Day 16        (4.2) 17      (4.2) 18 19       (4.4) 20 21 22       (4.4) 23    (4.5)/(4.6) 24     (4.6) 25 26       (4.7)Quiz 3 27 28 1         (4.7) 2         (4.7) 3      (4.9) 4 5  (4.9)/(4.10) 6 7 8        (7.2) 9        (7.2) 10      (7.3) 11 12       (7.3)Quiz 4 13 14 15     (7.4) 16      (7.4) 17Review 18 19Midterm 2 20 21 22    (7.6) 23     (7.8) 24  (7.8) 25 26  (9.1-9.3) 27 28 29 (9.1-9.3) 30  (9.1-9.3) 31  (9.4)Quiz 5 1 2 Univer. Day 3 4 5 (9.4) 6          (9.5) 7         (9.5) 8 9       (9.6) 10 11 12     (9.8) 13        (9.8) 14      (9.7) 15 16       (9.7) Quiz 6 17 18 19     extra 20      extra 21      extra 22 23     extra 24 25 26Review 27Review 28 Final 11am 29 30 no class 1

### Suggested Homework and Syllabus

 1.1 1,2,5,7,9,11 Dependent/independent variables, linear ODE 1.2 1a,2a,3,5,7,9,11,21,23,27,29a Solutions, Existence, Initial Value Problem 1.3 not covered Direction Fields 1.4 not covered Euler's Method 2.1 none Motion of a Falling Body 2.2 1,2,5,7,9,11,17,18,19,23,25,27,35,37 1rst Order Separable 2.3 2,3,4,7,9,13,15,17,18,19,22,37 1rst Order Linear 2.4 1,3,5 (solve as well),11,13,22,25,26 1rst Order Exact 2.5 not covered 1rst Order Special Integrating Factors 2.6 5,7,9,11 (implicit),15,21,23,25 1rst Order Homogeneous and Bernoulli only Suggested HW below is still tentative. 3.1 none Mathematical Modelling 3.2 1,3,7 Mixing models (only) 3.3 not covered Heating and Cooling Problems 3.4 1,5,24(hard) Newtonian Mechanics 3.5 not covered Electrical Circuits 3.6 not covered Improved Euler Methods 3.7 not covered Higher Order Numerical Methods Midterm 1 Content Summary Below REVIEW MATERIAL WILL BE POSTED IN D2L 4.1 none Introductory 2nd Order Models 4.2 1,5,9,13,19,27,31,37(r=1 root), 39 (r=2), 43 Homogeneous IVP, existence, Real Roots Case 4.3 1,3,5,9,11,13,19(r=1),21,25,29b (r=2),29c Homogenous, Complex Roots Case 4.4 9,11,13,15,17,23,25 (ugly),33 Nonhomogeneous: Undetermined Coeff. 4.5 3,7,17,19,23,25,27,33 (trig ident for cos^3),35 Nonhomogeneous: General solutions 4.6 1,3,5,7,11,13,17(longish) Variation of Parameters 4.7 9,11,13,15,17,19, Reduction of Order: 41,43,45 Cauchy-Euler equations, Reduction of Order 4.8 not covered Qualitative theory 4.9 1,7,9,11 Mechanical Vibrations 4.10 Covered but not on exam Mechanical Vibrations: Forced Midterm 2 Chapter 4 on HW material assigned REVIEW MATERIAL IS NOW POSTED IN D2L 5 Time permitting at end of course Phase Plane, Numerical 6 Time permitting at end of course General Theory of Linear Equations 7.2 3,5,9,11,13,15,17 Laplace Transform Definition 7.3 1,3,5,7,9,13,25,31 Laplace Transform Properties 7.4 1,3,7,9,21,23,25 (last 3 are nastier),33,35 Laplace Transform Inverse 7.5 1,3,7(nasty),11 (set y(t)=w(t-2)),15,17,19,35 Laplace Transform Initial Value Problems 7.6 TBA Laplace Transform Discontinuous Functions 7.7 not covered Laplace Transform of Periodic Functions 7.8 1,2,3,5,7,9,13 Laplace Transform Convolution Theorem 7.9 TBA Laplace Transform - delta function 7.10 not covered Laplace Transform - Systems of Equations 8 not covered Series Approximations and Solutions 9.1 1,3,5,8,11 Differential Equations as Systems 9.2 none Linear Algebraic Equations Gaussian Elimination 9.3 1,3,5,7b,7c,8,9,17,21,27,31,33,35,37,39 Matrix algebra and Calculus 9.4 1,3,5,9,13,15,19, 28!! Linear Systems - Normal Form 9.5 1,3,5,7,11,19,21,31!! Linear Systems - Constant Coefficient (Real Case) 9.6 1, 3 (given lamba=1),5,13a Linear Systems - Constant Coefficient (Complex Case) 9.7 11,13,21a Linear Systems - Variation of Parameters 9.8 Review problems. Linear Systems - Generalized eigenvectors and Repeated eigenvalues. Final: Wed. April 28, 11:00-11:50am            Lews Hall 304 (regular class) Chapter  9 : description below Review problems posted in D2L

### Exam and Quiz Content Descriptions:

 Quiz 1 1.1,1.2,2.2 Basic definitions (linear, order,......), explicit and implicit solutions of differential equations, verifying y(x) is a solution, finding the ODE for implicit solutions, separable equations and solving Initial Value Problems (IVP). There will NOT be anything on the falling body problem in (2.1) nor the Taylor series method in Chapter 1. Quiz 2 2.3,2.4,2.6 know separable, linear, exact, homogenous, Bernoulli definitions  and solution techniques. One question will be a chart where you decide if an exact is linear, homogenous, separable, Bernoulli. Three questions will be straight up solutions of first order ODE's of the aforementioned types. Quiz 3 4.2,4.3,4.4,4.5 Finding homogeneous solutions to all second order equations and possibly to a simple third order equation which is easily factorable. Finding particular solutions using Method of Undetermined coefficients. Lastly, finding general solutions (yh + yp) and solving IVPs. Quiz 4 4.6,4.7 Homogenous Cauchy-Euler equations and initial value problems. Variation of Parameters to find particular solutions given homogeneous solutions (you may need to compute the Wronskian). Reduction of Order to find second homogeneous solution and hence general homogeneous solution. NO questions on mechanical vibrations or Laplace transforms. Quiz 5 7.2,7.3,7.4,7.5,7.8 Laplace transform definition, finding transforms and inverse transforms, solving initial value problems, convolution theorem. There won't be anything from 7.6, 7.7, 7.9 of the text. You are allowed the Laplace Transform Summary sheet posted in D2L. Quiz 6 9.4,9.5,9.6,9.8 Fundamental matrices, general solns and solving Initial Value Problems. For the 2 by 2 matrix A, finding a general solution of x'=Ax where A has i) real distinct eigenvalues, complex eigenvalues and real repeated eigenvalues

Midterm 1

The exam will cover material from the following sections of the textbook:

1. Section 1.1  ODE definitions and theory
2. Section 1.2  IVP explicit/implicit solutions, existence uniqueness
3. Section 2.2  Separable Equations
4. Section 2.3  Linear Equations
5. Section 2.4  Exact Equations
6. Section 2.6  Homogeneous Equations and Bernoulli Equations only
7. Section 3.2  Mixing Problems (no population problems)
8. Section 3.4  Newtonian Mechanics - falling bodies, friction, rockets

Notes:

• You will have to solve a separable, linear, exact, homogeneous and Bernoulli equation. This forms the bulk of the exam ( about 70%)
• There will be an application problem: Only a mixing problem  (15%)
• There will be no questions on other applications: Newtonian Mechanics, Circuits,...
• One question will require you to categorize types of differential equations (15%).
• The sample problems posted in D2L are a good indication of the difficulty level of the problems.
•

Midterm 2  Content Description

The exam will cover material from the following sections of the textbook: 4.2-4.7, 4.9

1. Constant Coefficient 2nd order homogeneous yh(t)
2. Constant Coefficient 3rd order homogeneous yh(t) with one solution known (see review sheet)
3. Constant Coefficient 2nd order: Undetermined Coefficients Method for yp(t)
4. General Solutions y(t)=yh(t) + yp(t), Initial Value Problems, Wronskian for independence
5. Cauchy Euler 2nd Order homogeneous yh(t)
6. Variation of Parameter Method for yp(t) - standard form.
7. Reduction of order: homogeneous solution y2(t) from given homogeneous y1(t)
8. Mechanical Vibrations: Amplitude Phase Form y= A sin(wt+phi) for no friction case

Notes:

• There will be an amplitude-phase problem (10-15%). In fact, there will be a question from each point 1-8 above with the sole possible exception of 2.
• The sample review problems posted in D2L are a good indication of the difficulty level of the problems but this  sheet has only one amplitude-phase problem.
• Note: Undetermined coefficients is ONLY for L(y)=ay''+by'+cy=f and not L(y)=ax2y''+bxy'+cy=f
• There will be no Laplace Transform questions

Final: Content description

Wednesday April 28 - 11:00-11:50am in Lewis Hall 304 (regular class location)

Material from sections 9.4-9.8 of the textbook

#### Topics Covered

• Systems: Independence, Wronskian, Fundamental Matrix X(t)
• Systems: General Solution for homogeneous/nonhomogeneous systems
• Systems: Solving Initial Value Problems using fundamental matrix X(t)
• Systems: Constant A (2x2): real distinct eigenvalues
• Systems: Constant A (2x2): real repeated eigenvalues
• Systems: Constant A (2x2): complex eigenvalue
• Systems: Variation of Parameters

Updated on: 01/05/2021.